Asymptotic of the dissipative eigenvalues of Maxwell's equations

TL;DR

This paper analyzes the asymptotic distribution of eigenvalues of Maxwell's equations with dissipative boundary conditions in an unbounded domain, establishing a Weyl law for their counting function near the negative real axis.

Contribution

It provides a Weyl formula for the eigenvalue counting function of Maxwell's equations with dissipative boundary conditions when the boundary parameter varies.

Findings

Eigenvalues follow a Weyl law asymptotic distribution.

Established asymptotic behavior near the negative real axis.

Extended understanding of dissipative Maxwell eigenvalues.

Abstract

Let $\Omega = \mathbb R^3 \setminus \bar{K}$, where $K$ is an open bounded domain with smooth boundary $\Gamma$. Let $V(t) = e^{tG_b},\: t \geq 0,$ be the semigroup related to Maxwell's equations in $\Omega$ with dissipative boundary condition $\nu \wedge (\nu \wedge E)+ \gamma(x) (\nu \wedge H) = 0, \gamma(x) > 0, \forall x \in \Gamma.$ We study the case when $\gamma(x) \neq 1, \: \forall x \in \Gamma,$ and we establish a Weyl formula for the counting function of the eigenvalues of $G_b$ in a polynomial neighbourhood of the negative real axis.